Lower semicontinuity of integral functionals

Author:

Berkovitz Leonard D.

Abstract

It is shown that the integral functional I ( y , z ) = G f ( t , y ( t ) , z ( t ) ) d μ I(y,z) = {\smallint _G}f(t,y(t),z(t))d\mu is lower semicontinuous on its domain with respect to the joint strong convergence of y k y {y_k} \to y in L p ( G ) {L_p}(G) and the weak convergence of z k z {z_k} \to z in L p ( G ) {L_p}(G) , where 1 p 1 \leq p \leq \infty and 1 q 1 \leq q \leq \infty , under the following conditions. The function f : ( t , x , w ) f ( t , x , w ) f:(t,x,w) \to f(t,x,w) is measurable in t for fixed (x, w), is continuous in (x, w) for a.e. t, and is convex in w for fixed (t, x).

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference9 articles.

1. Existence theorems in problems of optimal control;Berkovitz, Leonard D.;Studia Math.,1972

2. Existence and lower closure theorems for abstract control problems;Berkovitz, Leonard D.;SIAM J. Control,1974

3. Remarks on the direct method of the calculus of variations;Browder, Felix E.;Arch. Rational Mech. Anal.,1965

4. Semicontinuity of multiple integrals in ordinary form;Fichera, Gaetano;Arch. Rational Mech. Anal.,1964

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